Question

The area bounded by the parabolas $y=4x^2,y=\frac{x^2}{9}$ and the line $y=2$ is

• A. $\frac{5\sqrt{2}}{3}$ sq. units
• B. $\frac{10\sqrt{2}}{3}$ sq. units
• C. $\frac{15\sqrt{2}}{3}$ sq. units
• D. $\frac{20\sqrt{2}}{3}$ sq. units

Answer 1

Answer: (D)

$y=4x^2...(i)$
$y=\frac{x^2}{4}...(ii)$
$\therefore A=\underset{0}{\overset{2}{\int }}\left[\frac{\sqrt{y}}{2}-3\sqrt{y}\right]dy$
$=\left(\frac{1}{2}-3\right)\underset{0}{\overset{2}{\int }}\sqrt{y}dy$
$=\left(\frac{-5}{2}\right)\cdot \frac{2}{3}{\left[y^{\frac{3}{2}}\right]}_0^2$
$=-\frac{5}{3}(2\sqrt{2}-0)=\left|\frac{-10\sqrt{2}}{3}\right|$
$=\frac{10\sqrt{2}}{3}$
$\Rightarrow$ Area of bounded Area $=2A=\frac{20\sqrt{2}}{3}$